Question

Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = 25x and y = x2 25 . Find V by slicing.

Answer 1

Answer:

$volume\ = \pi \frac{625^2}{6}$

Step-by-step explanation:

See the attached figure.

y₁ = 25x  and  y₂ =x²

The intersection between y₁ and y₂

25x = x²

x² - 25x = 0

x(x-25) = 0

x = 0 or x =25

y = 0 or y =25² = 625

The points of intersection (0,0) and (25,625)

To find the volume of the solid obtained by rotating about the y-axis the region bounded by y₁ and y₂

y₁ = 25x ⇒ x₁ = y/25 ⇒ x₁² = y²/625

y₂ =x² ⇒ x₂ = √y ⇒ x₂² = y

v = ∫A(y) dy = π ∫ (x₂² - x₁²) dy

∴ V =

$\pi \int\limits^{625}_0 {y-\frac{y^2}{625} } \, dy =\pi( \frac{y^2}{2} -\frac{y^3}{3*625} ) =\pi (\frac{625^2}{2} -\frac{625^3}{3*625}) =\pi ( \frac{625^2}{2}-\frac{625^2}{3}) =\pi \frac{625^2}{6}$